GEOPHYSICAL RESEARCH LETTERS, VOL. 40, 3074–3079, doi:10.1002/grl.50584, 2013
Field measurements and scaling of ocean surface
wave-breaking statistics
Peter Sutherland1 and W. Kendall Melville1
Received 30 April 2013; revised 21 May 2013; accepted 22 May 2013; published 17 June 2013.
[1] Deep-water breaking waves provide a mechanism for
mass, momentum, and energy transfer between the atmosphere and ocean. Microscale breaking is particularly important, but notoriously difficult to measure in the field. In this
paper, measurements from a new technique, using a stereo
pair of long-wave infrared cameras to reconstruct the sea
surface shape and velocity field, are presented. Breakers are
detected using an image texture-based algorithm and then
tracked on the reconstructed surface. These waves range
from large air-entraining breakers to microbreakers that are
undetectable by traditional visible video-based techniques.
This allows measurements of breaker length distributions,
ƒ(c), that extend to velocities near the gravity-capillary
transition. These distributions are compared with measurements from the literature and from visible video imagery.
A nondimensional scaling is proposed which collapses
ƒ(c). Finally, estimates of energy dissipation and stress
based on ƒ(c) are found to agree well with wave energy
dissipation and wind stress models. Citation: Sutherland,
The fraction of surface area turned over by breaking fronts
per unit time is the first moment of ƒ(c) [Phillips, 1985]
Z
R=
(1)
which is related to heat and gas transfer between the ocean
and the atmosphere [Jessup et al., 1997].
[4] Phillips [1985] showed that the average rate of gravity
wave energy loss per unit area by breakers with velocities
between c and c + dc can be written,
(c)dc = bg–1 w c5 ƒ(c)dc.
(2)
where g is the gravitational acceleration, w is the water density, c is the breaker front velocity, and b is the dimensionless
breaking parameter.
[5] Integrating over all breaker speeds c, the fifth moment
of ƒ(c) gives the total gravity wave energy dissipated by
breaking waves per unit area of ocean surface,
P., and W. K. Melville (2013), Field measurements and scaling
of ocean surface wave-breaking statistics, Geophys. Res. Lett., 40,
3074–3079, doi:10.1002/grl.50584.
FE =
w
g
Z
bc5 ƒ(c)dc.
(3)
Similarly, the fourth moment of ƒ(c) yields the momentum
flux from the wave field into the upper ocean,
1. Introduction
[2] The air-sea interface has a profound effect on weather
and climate, with deep-water breaking waves playing an
important role [Banner and Peregrine, 1993; Melville,
1996; Duncan, 2001]. Energy dissipation by breakers limits
growth of the surface wave field and provides a mechanism
for momentum transfer from waves to currents. Bubbles and
aerosols produced by breaking support gas and heat transfer between the ocean and atmosphere. Breaking statistics
are very difficult to simulate in the laboratory or to model
numerically and so field measurements are necessary for
progress.
[3] Phillips [1985] defined a distribution of breaker front
length ƒ(c) per unit area of sea surface per unit increment of
breaking front velocity c(c, ), where c and are the speed
and direction of breaker front propagation, respectively. The
moments of ƒ(c) have important physical interpretations.
Additional supporting information may be found in the online version
of this article.
1
Scripps Institution of Oceanography, University of California, San
Diego, La Jolla, California, USA.
Corresponding author: P. Sutherland, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0213, USA.
([email protected])
©2013. American Geophysical Union. All Rights Reserved.
0094-8276/13/10.1002/grl.50584
cƒ(c)dc,
Fm =
w
g
Z
bc3 cƒ(c)dc.
(4)
Phillips [1985] used a constant value for b and, by assuming
an equilibrium wave spectrum, predicted that ƒ(c) / c–6 .
More recent laboratory work using focusing wave packets
[Melville, 1994; Banner and Peirson, 2007; Drazen et al.,
2008] has shown that b varies over at least three orders of
magnitude. For plunging breakers, Drazen et al. [2008] predicted that b is proportional to S5/2 , where S is the linear
focusing slope at breaking. Romero et al. [2012] showed that
b = 0.4(S – 0.08)5/2 fits all available laboratory data, from
incipient to plunging breaking, and used it to create a semiempirical spectral model of the breaking parameter in the
field,
5/2
b(k) = A1 (B(k)1/2 – B1/2
T ) .
(5)
Here B(k) is the azimuth-integrated saturation spectrum,
with the wave number k related to c using the linear dispersion relation for deep-water gravity waves. BT and A1
are constants that were determined by balancing wave field
dissipation [Romero and Melville, 2010] with the dissipation from breaking calculated with equation (2), using
the field measurements of ƒ(c) by Kleiss and Melville
[2010]. This model for b(k) was used by Romero et al.
[2012] to predict a form of ƒ(c), at low c values, that
extended roughly from the peak of the field measurements of
3074
SUTHERLAND AND MELVILLE: SURFACE WAVE-BREAKING STATISTICS
102
100
σc
90
80
70
10−2
60
10−4
cp/u*
Λ(c) [m−2s]
100
50
J.&P. 2005
M.&M. 2002
G. et al. 2008
P. et al. 2001
K.&M. 2010
R. et al. 2012
−6
10
10−8
10−1
40
30
20
100
101
c [m/s]
Figure 1. Measured ƒ(c) compared with the literature. Lines colored by wave age are from this work. Thick black lines
are from the laboratory measurements of Jessup and Phadnis [2005]. Field measurements are from Melville and Matusov
[2002] lines with small dots, Gemmrich et al. [2008] lines with squares, Phillips et al. [2001] lines with triangles, and Kleiss
and Melville [2010] solid gray lines. The black and white dashed line is the modeled extrapolation by Romero et al. [2012]
of the measurements of Kleiss and Melville [2010]. Horizontal bars near the top indicate standard deviation of velocity error
calculated using synthetic data.
Kleiss and Melville [2010] to the peak of the laboratory
measurements of Jessup and Phadnis [2005].
[6] Several studies have undertaken direct measurements
of ƒ(c) in the field, as summarized in Figure 1. Although
early work by Phillips et al. [2001] used radar backscatter,
all field measurements since then have used video imagery
of the sea surface. Melville and Matusov [2002] and Kleiss
and Melville [2010, 2011] used airborne imagery, while
Gemmrich et al. [2008] and Thomson et al. [2009] used
platform-based measurements.
[7] Detection of breakers in video imagery relies on
whitecaps being much brighter than the surrounding ocean
due to the scattering of light by entrained bubbles and foam.
Microscale breakers do not entrain air, and so are essentially undetectable by visible imagery. They do, however,
mix the cool skin layer with warmer water below, so their
warm actively breaking front and wake can be detected in
IR imagery [Zappa et al., 2001]. Jessup and Phadnis [2005]
used IR video to image wind-generated microscale breakers in the laboratory. Their measured ƒ(c) showed a much
higher density of breakers at lower speeds than previous
visible video-based fieldwork.
[8] The primary goal of this work is to include microscale
breakers in field measurements of ƒ(c). This has been
achieved by using stereo IR measurements to capture breaking waves in the open sea. To the authors’ knowledge,
it represents the first successful inclusion of microscale
breakers in field measurements of ƒ(c).
2. Experiment and Methods
[9] The data described here were collected during three
deployments of Research Platform FLIP in the Pacific
Ocean in 2009 and 2010. Radiance in a Dynamic Ocean
2009, RaDyO 2009, [Dickey et al., 2012], was a 12-day
deployment that started 120 km south of the Island of
Hawai’i with FLIP drifting west at approximately 35 cm/s
(0.7 knots) for approximately 330 km in typical trade
winds. The Office of Naval Research-sponsored High
Resolution Air-Sea Interaction Departmental Research
Initiative consisted of two experiments: The main experiment, HiRes, 2010 was a 14 day deployment with FLIP
moored approximately 25 km off the coast of Northern
California (38ı 200 N, 123ı 260 W) in strong northwesterly
winds. The second experiment, SoCal 2010, took place over
2 days in the Southern California Bight in much milder
conditions. Between the three experiments, 70 20-minute
records were analyzed with wind speeds, U10 , of 1.6 to
16 m/s, significant wave heights, Hs , of 0.7 to 4.7 m, and
wave ages, cp /u* , of 16 to 150 (where cp is the phase speed
of waves at the spectral peak and u* is the atmospheric
friction velocity).
[10] The primary instrumentation was a pair of FLIR
SC6000 long-wave infrared (8–9.2 m) video cameras.
The cameras were mounted 3 m apart on a horizontal spar
at the end of one of FLIP’s booms. The cameras were
angled slightly toward each other so that they shared
the same field of view on the sea surface and angled 20ı
from vertical away from FLIP in order to reduce reflections from FLIP’s superstructure and booms. The colocated
field of view was approximately 4 3 m, and the image
size of 640 512 pixels resulted in a nominal resolution
of approximately 6 mm (which changed depending on the
instantaneous boom and surface displacement). IR video
was captured at 40 Hz (sub-sampled at 20 Hz) for the first
20 min of every hour.
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SUTHERLAND AND MELVILLE: SURFACE WAVE-BREAKING STATISTICS
15.4
0.5
15.2
15.2
15
15
14.8
14.8
14.6
2 m/s
Y, North [m]
Y, North [m]
15.4
0°C
14.4
14.2
14.6
14.4
14.2
14
14
(a)
13.8
17.6
17.8
(b)
18
18.2
18.4
-0.5
13.8
17.6
X, East [m]
17.8
18
18.2
18.4
X, East [m]
Figure 2. Snapshot of a breaker detected in co-located visible and IR imagery, taken during SoCal 2010 experiment, December 6, 2010, 22:02:32.75 [UTC]. (a) IR temperature image with 20-minute mean removed. Black arrows
indicate breaking front velocities [m/s] (scale given in image), green dashed line indicates the boundary of the cutout in the
visible image (white area in Figure 2b). (b) Visible (black and white) image. Blue arrows indicate breaking front velocities
detected in visible imagery, orange arrows indicate velocities detected in IR. The white area in the lower left was removed
from analysis as it contained a subsurface instrument which was visible through the clear ocean water and thus affected
breaker detection in visible imagery. Wide field of view versions of these images are available in Supplementary Figures 4
and 5.
[11] Over the duration of these experiments, the sea surface contained a wide variety of thermal structures that
were detectable by the IR cameras (i.e., temperature differences greater than the detectable minimum of approximately
25 mK). These structures included actively breaking waves,
remnants of past breakers, and the surface signatures of convective and other turbulence. By matching these features in
both stereo images, it was possible to triangulate the location
of the features in 3-D. The use of IR imagery in the laboratory [Hilsenstein, 2005] has shown it to be an effective
way of eliminating the principal difficulties of using visible stereo on a water surface, namely water penetration and
specular reflection [Jähne et al., 1994]. By assuming that
these same temperature structures behaved as passive tracers
over the time scale of the video frame separation (0.05 s),
and tracking them between consecutive frames, it was possible to derive the water surface velocity. Further details
of the stereo thermal structure particle imaging velocimetry
(PIV) processing are available in the supporting information.
This thermal structure PIV technique has been used by other
authors [Garbe et al., 2003; Veron et al., 2008; Rocholz
et al., 2010] to measure surface velocity with a single IR
camera (under the assumption of a flat sea surface), but to
the best of our knowledge, has never before been combined
with stereo IR imaging to reconstruct the 3-D velocity at the
sea surface.
[12] In addition to stereo IR imagery, a variety of
supporting measurements were taken. Colocated stereo
visible video was recorded for part of the SoCal 2010 experiment. Winds and atmospheric fluxes were measured with
a CSAT3 3-D sonic anemometer eddy flux system (Campbell Scientific) mounted directly over the IR cameras’ fields
of view.
2.1. Breaker Detection
[13] In this work, a new technique was developed for
breaker detection. Temperature structure in the actively
breaking region and turbulent wake of a breaker is less
uniform than that in the background surface field. One
measure of non-uniformity in an image is “entropy,” which
is frequently used in the computer vision and synthetic
aperture radar communities to quantify image texture (e.g.,
Gonzalez et al. [2009], Holmes et al. [1984]). Local entropy
was used to detect breaking waves in infrared imagery. The
supporting information contains a description of the technique and the method used to calculate ƒ(c). Four 10 min
periods of visible video data, corresponding to the first half
of their associated IR sampling periods, were also analyzed
for comparison. The technique used for visible imagery was
based on that developed by Kleiss and Melville [2011] and
is described in the supporting information. An example of
a breaker detected in both visible and IR imagery can be
seen in Figure 2 and Figures S4 and S5. Note that the
main bubble-entraining “whitecap” is obvious in both visible and IR images, with both measurements finding the
actively breaking front in the same location with very similar
velocities. However, the IR image shows the main breaker
extending in the y direction well beyond the visible whitecap
boundary and also shows several other smaller breakers that
are completely undetected in the visible image.
3. Results
[14] The ƒ(c) measurements taken here are shown in
Figure 1. They show, at the higher speeds, a similar
functional dependence on c when compared to other examples from the literature. What sets the measurements in this
3076
SUTHERLAND AND MELVILLE: SURFACE WAVE-BREAKING STATISTICS
60
(a)
100
(b)
100
55
10−4
45
10−2
40
cp/u*
10−2
Λ(c) [m−2s]
Λ(c) [m−2s]
50
35
10−4
30
10−6
−1
10
0
10
Zappa et al. 2012
−6
c
Stereo IR
10−6
Stereo IR
Stereo Visible
1
−1
10
10
c [m/s]
0
10
25
20
1
10
c [m/s]
Figure 3. Comparison of ƒ(c) measured using visible and IR video. Color corresponds to wave age. (a) Distributions from
concurrent, colocated IR video (solid lines) and visible video (dashed lines) taken during the SoCal 2010 experiment. (b)
ƒ(c) measured during the RaDyO 2009 experiment using stereo IR (colored lines) compared with ƒ(c) measured during the
same experiment by Zappa et al. [2012] using visible imagery (colored circles). The solid black line is the c–6 dependence
predicted by Phillips [1985] as shown in Zappa et al. [2012].
work apart is their low-speed behavior, extending the field
measurements of Melville and Matusov [2002], Gemmrich
et al. [2008], Kleiss and Melville [2010], and others. They
also appear to have a similar peak speed to the laboratory
measurements of Jessup and Phadnis [2005] though that
agreement may be coincidental.
[15] A recurring question in the literature is whether
or not ƒ(c) should roll-off below some breaking speed c
[Gemmrich et al., 2008; Kleiss and Melville, 2011]. As can
be seen in Figure 3, including microscale breakers has a very
large effect on the level of ƒ(c) at low c. Both visible and IR
measurements capture breaking fronts well for breakers with
speeds above 2–3 m/s. Below that speed, visible measurements fail, likely due to the lack of air entrainment. The IR
measurements presented here do not display a similar rolloff until c reaches between 10 and 80 cm/s. The standard
deviation of velocity error was calculated to be approximately 20 cm/s (see supporting information), which likely
broadens the peak in the ƒ(c) distributions. Further supporting the general form of these low c results is their similarity
to the modeling work of Romero et al. [2012]. By using a
theoretical extrapolated wave saturation spectrum, they predicted that ƒ(c) given by Kleiss and Melville [2010] should
increase from the measured peak as c decreases, as shown in
Figure 1.
p
Here gHs is the speed attained in a ballistic trajectory
from a height of Hs /2, cp /u* is the wave age, gHs /c2p is
the wave steepness, gX/c2p is the dimensionless fetch, and
Bo is the Bond number. The spectral Bond number, Bo =
(w – a )c4 /g is greater than 10 for c > 29 cm/s, implying that surface tension effects are negligible for speeds
larger than this. Our RMS error in c is ˙20 cm/s, so surface tension effects are negligible over most of the range
4. Scaling ƒ(c)
[16] A dimensional analysis of the dependence of ƒ(c)
on the other variables and parameters leads to an improved
understanding of the breaking process. ƒ(c) can be written
as ƒ(c; a , w , u* , g, cp , Hs , , X), where a is the air density,
u* is the atmospheric friction velocity, cp is the phase speed
of waves at the peak of the wind-wave spectrum, Hs is the
significant wave height, is the surface tension, and X is the
wave fetch. Dimensional analysis then yields
!
c
a cp gHs gX
ƒ(c)c3p g–1 = f p
, , , 2 , 2 , Bo ,
gHs w u* cp cp
(6)
Figure 4. Nondimensional breaking length distribution.
Distributions have been binned by wave age with corresponding colors. Solid lines (
) are measurements taken
)
in this work using stereo IR imagery, dash-dotted (
lines are from visible imagery in this work, and dashed lines
(
) are from the airborne measurements of Kleiss and
Melville [2010]. Scaling uses ballistic velocity, steepness,
and wave age dependence from equation (9).
3077
SUTHERLAND AND MELVILLE: SURFACE WAVE-BREAKING STATISTICS
ρwg−1∫ bc5Λ(c)dc [W/m2]
!
(7)
.
Assigning the wave age dependence to the ordinate and the
steepness dependence to the abscissa, and assuming a power
law dependence on both wave age and steepness, gives
c3p
g
cp
u*
˛
c
=f p
gHs
gHs
c2p
100
(8)
Applying this scaling to all the stereo IR ƒ(c) distributions
of Figure 1 and then varying ˛ and to collapse the curves
(by minimizing a squared-difference cost function) gave values of ˛ = 0.5 and = 0.1. Using those values, it is then
possible to rewrite equation (8) as
ƒ(c)
c3p
g
cp
u*
0.5
0
= f@p
c
gHs
gHs
c2p
!0.1 1
A = f(Oc),
40
RaDyO 2009
HiRes 2010
SoCal 2010
1:1
10−2
−1
10
0
10
20
1
10
−ρwg∫ Sds(k)dk [W/m2]
! !
.
60
10−1
10−3 −2
10
ρwg−1 ∫ bc3 cΛ(c)dc [N/m2]
ƒ(c)
80
100
100
(b)
80
10−1
60
10−2
40
(9)
20
10−3 −3
10
where cO is the scaled breaker front speed. This form of cO
suggests that the relevant
p velocity for scaling breaker front
length distributions is gHs , with a weak dependence on the
peak wave steepness, Hs kp = gHs /c2p . The ballistic velocity,
p
gHs , has been used by previous authors for scaling breaking processes. Drazen et al. [2008] used it along with inertial
scaling of dissipation to obtain the S5/2 slope dependence of
the breaking parameter that Romero et al. [2012] used to
deduce equation (5).
[17] Figure 4 shows ƒ(c) measurements from stereo IR
and stereo visible, as well as the airborne visible measurements of Kleiss and Melville [2010], scaled using
equation (9). For speeds higher than the distribution peak,
found between cO = 0.06 and 0.1, this scaling collapses all
stereo IR measurements to a narrow curve. For larger speeds,
that curve approaches 0.05 cO –6 , exhibiting the power law
dependence predicted by Phillips [1985]. One curve, corresponding to the range cp /u* = 10 – 20, fails to collapse as
well as the others. Those data were taken during the HiRes
2010 experiment during which strong conditions and low
air-water temperature differences reduced contrast in the IR
imagery and made detection of breakers less reliable than in
the other experiments. Thus, it is possible that the premature
roll-off at low and high speeds of the cp /u* = 10 – 20 curve
is due to measurement limitations rather than the breakers
themselves.
[18] While the cO –6 regions of the distributions measured
using visible video show reasonable agreement with the
high-speed region of the stereo IR distributions, the regions
at velocities below the peaks of the visible video data, at
approximately cO = 0.8, do not. This supports the hypothesis that the peaks found in ƒ(c) distributions measured using
visible video are a result of their inability to measure breakers that do not entrain air. It also suggests that cO = 0.8
may be an important dynamic transition for air entrainment
by breaking.
cp/u*
cp gHs
c
ƒ(c)c3p g–1 = f p
; ,
gHs u* c2p
100
(a)
cp/u*
101
of c and are neglected in this analysis. However, since the
ratio (w – a )/g remained effectively constant over all
experiments, these measurements had no significant Bond
number dependence. Neglecting the Bond number, assuming that a /w is approximately constant, and assuming that
in fetch-limited conditions X can be related to cp , Hs , u* , and
g, equation (6) can be simplified to
−2
10
10
−1
0
10
ρau*2 [N/m2]
Figure 5. (a) Dissipation by breaking (ordinate) compared with modeled wave field dissipation (abscissa).
(b) Momentum flux from waves to currents due to wave
breaking (ordinate) plotted against wind stress (abscissa).
Color shows wave age and solid line indicates 1:1
correspondence.
[19] The ability to nondimensionalize ƒ(c) over the wide
range of wind and wave conditions in these experiments is
an important step toward parameterization of wave-breaking
statistics. Peak wave speed, friction velocity, and significant
wave height are much more commonly and easily measured
than ƒ(c).
5. Discussion
[20] Until this section, results have been presented with
no assumptions made regarding the relationship between
the speed of the breaking front c and the phase speed of
the underlying wave cw . However, Phillips [1985] and others have assumed that cw = ac, where a is a constant of
proportionality approximately equal to 1. In this discussion
section, we explore the dynamics assuming cw = c. Using
equation (3), with the breaking parameter b(k) as modeled
in equation (5) and mapped to b(c) using the linear dispersion relation for surface gravity waves, it is possible to
estimate the total integrated dissipation by wave breaking.
Since Romero et al. [2012] developed the expression for
b(c) in the context of surface gravity waves, b(c) can only
be used for speeds where the inclusion of surface tension
has little effect. For the following analysis, the restriction of
c > 0.29 m/s was applied, corresponding to wave num-
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SUTHERLAND AND MELVILLE: SURFACE WAVE-BREAKING STATISTICS
bers at which the spectral Bond number (section 4) is greater
than 10.
[21] Figure 5a shows dissipation from measured breaking plotted against a model of the integrated spectral wave
dissipation. The spectral wave dissipation model, Sds , is
from Romero and Melville [2010] and based on Alves and
Banner [2003]. Wind input was calculated using the formulation of Janssen [1991]. Visually, agreement is good.
For wave ages below cp /u* = 50, the relationship between
dissipation by breaking and modeled dissipation is approximately linear. For very old waves, the linear relationship
breaks down, suggesting that either breaking is not the dominant mechanism for wave dissipation, or the dissipation
model fails, at high wave ages. Figure 5b shows that similarly momentum flux from breaking approaches equivalence
with wind stress at low wave ages. The scatter in both
the dissipation and stress is well within the scatter of the
data used by Romero et al. [2012] to parameterize b(k) in
their model.
[22] Acknowledgments. These measurements would not have been
possible without the exceptional engineering and field support of Luc
Lenain and Nick Statom. We also thank Tom Golfinos, Bill Gaines, and
the crew of R/P FLIP who always “went the extra mile.” We thank
Fabrice Veron and Zachary VanKirk for assisting with data collection
during the HiRes 2010 experiment and for providing an early version of
the stereo matching algorithm; Luc Lenain for providing useful video analysis software and much valuable advice; Laurent Grare for processing the
meteorological data for the HiRes 2010 experiment; Jessica Kleiss for providing airborne ƒ(c) measurements from GOTEX, and Leonel Romero
for many useful discussions and providing Matlab code for calculating
model wind input. We are grateful for the efforts of the two anonymous reviewers whose insights greatly improved this paper. This research
was conducted under grants to WKM from the Office of Naval Research
(HiRes and RaDyO DRIs) and the National Science Foundation (Physical
Oceanography).
[23] The Editor thanks two anonymous reviewers for their assistance in
evaluating this paper.
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